On the statistical properties of finite continued fractions

A. V. Ustinov

Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 186-211

Résumé

The article is devoted to the statistical properties of continued fractions for the numbers $a/b$, for $a$ and $b$ in the sector $a,b\ge1$, $a^2+b^2\le R^2$. Main result is asymptotic formula with two meaning terms for the value $$ N_x(R)=\sum_{a^2+b^2\le R^2\atop a,b\in\mathbb{N}}s_x(a/b), $$ where $s_x(a/b)=|\{j\in\{1,\ldots,s\}:[0;t_j,\ldots,t_s]\le x\}|$ is Gaussian statistic for the fraction $a/b=[t_0;t_1,\ldots,t_s]$.